Normalisers of Sylow subgroups in reflection groups

Let $W$ be a finite reflection group, $\ell$ a prime divisor of $|W|$ and $S_\ell$ a Sylow $\ell$-subgroup of $W$. It can be shown that there exists a unique parabolic subgroup $P$ minimally containing $S_\ell$. We prove that there exists a simple system of $W$ such that $P$ is a standard parabolic subgroup and $N_W(S_\ell)=N_{P}(S_\ell) \rtimes U,$ where $U$ is the setwise stabiliser of the simple roots of $P$. We also determine when an analogous decomposition of the normaliser of a Sylow $\ell$-subgroup exists when we replace $P$ by a reflection subgroup minimally containing $S_\ell$. Furthermore, it is shown that if $W$ is crystallographic and has no proper parabolic subgroup containing a Sylow $\ell$-subgroup of $W$, then there is a Sylow $\ell$-subgroup normalised by the Dynkin diagram automorphisms of $W$.Comment: 12 pages, 1 figure, 2 table